Extending this approach depends on our ability to use the Taylor expansion and the Taylor-Lagrange remainder in \(\mathbb {H}\) , however in
[1]} Chapter 4 (theorem 4.5) an extension to Banach spaces of the Taylor expansion is defined, and since \(R\)  is a real-valued function, we know that the mean value theorem is valid; thus, the existence of the Taylor-Lagrange remainder is guaranteed.
